Download From cardiac electrical activity to the ECG

From cardiac electrical
activity to the ECG
A finite element model
Master thesis
A.M.J Lenssen
535227
BMTE 08.26
Committee:
Prof. Dr. Ir. F.N. v.d. Vosse
Dr. Ir. P.H.M. Bovendeerd
Dr. Ir. G.J. Strijkers
Ir. N.H.L. Kuijpers
Drs. P.M. v. Dam (Medtronic, Arnhem)
Eindhoven University of Technology
Department: Biomedical Engineering
Division: Biomechanics and tissue engineering
Group: Cardiovascular Biomechanics
Eindhoven, May 2008
Abstract
In the group of Cardiovascular Biomechanics, finite element models of cardiac electromechanics are developed, that ultimately might be used as a tool to assist in diagnosis of cardiac
disease. In such a diagnosis, with the use of the model, clinically available data such as the
ECG of MR tagging images would be translated into an estimation of cardiac tissue properties. The current models lack the possibility to translate the ECG into a prediction of cardiac
electrical activity. As a first step towards this possibility, in the present thesis the goal is to
develop and implement a model to solve the forward relation between cardiac electrical activity and the ECG. First a few test problems were analyzed in a two dimensional domain, where
various descriptions for the cardiac source were considered. From this it was concluded that
modeling cardiac activity through electric potentials yielded a more robust estimation of the
potential field, than a model in which cardiac activity was modeled through electric current
sources. Next, the relation between cardiac electrical activity and the ECG was investigated
in a geometrically more realistic 3D model. The geometry of the left and right ventricle of a
dog heart was placed inside a stylized torso model. Activation times were determined from
a wave speed model, the Eikonal equation. Action potential dynamics were estimated and
coupled to the activation times. The potential distribution caused by cardiac potentials, was
calculated throughout the torso and body surface potentials were determined on the front of
the torso model. A 12 lead ECG was calculated and results were compared to model results
obtained by Miller et al. The body surface potential pattern and the ECG showed many
differences with the results of Miller et al. Therefore a few parameters that could influence
the body surface potentials were analyzed. Making the action potential duration heterogeneous throughout the myocardium was found to have an effect on the calculated T waves in
the ECG only. Changing the orientation of the heart with respect to the thorax affected the
body surface potentials only slightly. Since the variations did not yet lead to a more realistic
result, the parameters need more study. In conclusion, a model was developed that provides
the ability to calculate body surface potentials from a description of cardiac activation. To
obtain realistic results more parameter studies are needed.
ii
Contents
1 General introduction
1
1.1
Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1.2
Outline of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
2 Theoretical Background
5
2.1
Anatomy of the heart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
2.2
Electrophysiology of the heart . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
2.3
The electrocardiogram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
2.3.1
The 12 lead ECG
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
2.3.2
The ECG signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
3 Modeling electrical activity
3.1
3.2
11
The bidomain model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
3.1.1
The heart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12
3.1.2
The torso . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14
3.1.3
Summary of the bidomain approach . . . . . . . . . . . . . . . . . . .
14
Decoupling the problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15
3.2.1
Eikonal equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16
3.2.2
Common methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17
4 Two dimensional test problems
4.1
4.2
21
Dipole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21
4.1.1
Analytical description . . . . . . . . . . . . . . . . . . . . . . . . . . .
21
4.1.2
Numerical Implementation
. . . . . . . . . . . . . . . . . . . . . . . .
22
4.1.3
Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
24
4.1.4
Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
24
Dipole layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
25
iii
4.3
4.4
4.2.1
Analytical description . . . . . . . . . . . . . . . . . . . . . . . . . . .
25
4.2.2
Numerical implementation . . . . . . . . . . . . . . . . . . . . . . . . .
26
4.2.3
Simulations performed . . . . . . . . . . . . . . . . . . . . . . . . . . .
26
4.2.4
Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
27
4.2.5
Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
Volume source versus surface source . . . . . . . . . . . . . . . . . . . . . . .
30
4.3.1
Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
4.3.2
Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
General Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
5 From cardiac activation to body surface potentials
5.1
33
Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
5.1.1
Activation of the heart . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
5.1.2
Heart inside the torso . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
5.1.3
Determining the ECG . . . . . . . . . . . . . . . . . . . . . . . . . . .
36
5.2
Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
37
5.3
Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
38
6 Parameter study
6.1
6.2
45
Heterogeneous action potential duration . . . . . . . . . . . . . . . . . . . . .
45
6.1.1
Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
45
6.1.2
Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
45
6.1.3
Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
47
Orientation of the heart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
47
6.2.1
Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
47
6.2.2
Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
48
6.2.3
Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
49
7 General discussion
51
A Results of variations in orientation of the heart.
57
iv
Chapter 1
General introduction
1.1
Background
In the clinic, information about the functioning of the heart is mainly obtained by electrocardiography and various non invasive imaging techniques such as magnetic resonance tagging
(MRT). The ECG and the deformation pattern obtained with these clinical measurements
give information about the status of the myocardium and about abnormalities in the tissue, such as a conduction disorder or cardiac ischaemia. Diagnosing a patient involves the
translation of the ECG and the deformation pattern into a conclusion on the status of the myocardium. Traditional diagnosis is often based on clinical experience and not all information
available from measurements is used. Mathematical models of cardiac function can play a
role in studying the relation between clinical measurements and the status of the myocardium
and might be used to assist in diagnosis.
At the group of Cardiovascular Biomechanics, mechanical models of the heart have been studied extensively. Initially, the influence of parameter variations on local wall mechanics was
studied with models that are geometrically simple. With the use of clinical data about the
wall mechanics obtained with for instance MRT, these models are tuned to gradually make
them useful for diagnostics.
Cardiac muscle contractions are triggered by electrical stimulation of the cardiac muscle.
Therefore the electrical functioning of the heart is of influence on the mechanical characteristics. In the current models, the initiation of muscle contraction is regulated by an electrical
stimulation sequence which is calculated using a wave speed model [25]. Studies showed that
this model is capable of simulating left ventricular mechanics for hearts with normal and
abnormal pulse conduction[20]. However, there were still some differences with respect to
clinical measurements when comparing local deformation patterns. Furthermore, simulations
were performed to study whether the model is able to simulate cardiac wall strain for normal
activation and in case of a left bundle branch block (LBBB)[29]. The activation sequence was
adjusted to what was found in literature. From this simulations could be concluded that the
model is not yet capable of simulating physiologic strain patterns in a left ventricle with a
LBBB.
Information about the electrical status of the myocardium can be of additional value to the
mechanical models described. Since clinically cardiac activation cannot be measured directly,
the ECG is used as an indirect measure for the electric functioning of the heart. The ECG
1
1.2 Outline of the thesis
General introduction
represents the effect of distribution of potentials in the cardiac muscle on the body surface.
Since cardiac potentials are difficult to assess, mathematical models play an essential role in
studying the genesis of the ECG. Mathematical models used to calculate body surface potentials from a description of the cardiac activation sequence, are called forward models. In
the situation where cardiac activation is determined from potentials on the body surface, the
inverse problem is solved. To obtain more insight in the relation between cardiac electrical
activity and body surface potentials, the aim of this study is to develop a model to calculate
body surface potentials from cardiac electrical activity.
Over the years, many studies have been performed on the forward problem of electrocardiography. Since the first numerically solved forward model by Gelernter and Swihart in 1964
[18], the development in numerical methods and computer hardware have increased the complexity of recent mathematical models. Although the possibilities of computational power
have increased, there are still limits in the level of detail one is able to incorporate in the
model. Microscopic descriptions of myocardial tissue are used by many research groups [9, 17].
The level of detail in these models brings them closer to the physiological situation, however
modeling a complete heart built from blocks the size of a single cell can give problems with
respect to computation time and memory. Therefore, in modeling electrical activity of the
complete heart, more macroscopic approaches are more convenient.
The heart, the torso, and possible other conductive regions that play a role in the problem
are represented by either boundary surfaces or volumes. When the torso is represented as
a conducting volume, a volume method is used. A distinction can be made between models
where the cardiac wall volume is taken into account and approaches where the activation
of the heart is described only on the surface. If the model only comprises the heart and
torso boundary surfaces a surface method is used. In this situation, often a boundary element method (BEM) is used to represent the geometries numerically. The activation of
the heart is represented on the surface of the heart. Homogeneous and isotropic conducting
properties are assumed in each compartment to calculate body surface potentials. In many
cases, regions with different conductive properties inside the torso, for example the lungs,
are taken into account [3, 28]. The volume method needs more computational time than a
surface method because a larger amount of elements is needed, but a great advantage of the
use of volume elements is the ability to incorporate the different conducting properties of the
various regions, such as the lungs, that are present between the epicardium and the body
surface. Models of cardiac mechanics, like the models developed by Kerckhoffs et al. [25],
are generally implemented in finite element (FE) packages. To be able to eventually couple
electrical modeling with mechanical modeling, a model of cardiac electrical activity is most
valuable when solved with a FE model. To be able to incorporate anisotropic conduction
properties and to connect with mechanical models, the aim is to use a FE model to describe
the forward problem numerically.
1.2
Outline of the thesis
In chapter 2 the anatomy and (electro-)physiology of the heart will be briefly discussed, as
well as the concept of electrocardiography. Subsequently, the modeling of electrical activity
of the heart to the body surface will be discussed in chapter 3. In chapter 4 a few two
dimensional test problems are performed where various models are considered to represent
the cardiac source. A simplified three dimensional forward model is analyzed in chapter 5. In
2
1.2 Outline of the thesis
General introduction
chapter 6 the influence of different parameters on the solution determined with the model is
studied. Finally, in chapter 7 an overall discussion and conclusion of the study is presented.
3
1.2 Outline of the thesis
General introduction
4
Chapter 2
Theoretical Background
2.1
Anatomy of the heart
The heart consists of four chambers, the left and right ventricle and the left and right atrium.
In figure 2.1 a drawing of the anatomy of the heart is shown. The atria are separated from
the ventricles by the base, where the the four cardiac valves are situated; the tricuspid valve
between the right atrium and the right ventricle, the mitral valve between the left atrium and
the left ventricle, the aortic valve between the left ventricle and the aorta and the pulmonary
valve between the right ventricle and the pulmonary artery. The atria are relatively thin
walled cavities that function as a weak filling pump for the ventricles. The right and left
ventricle are separated by the septum. The walls of the ventricles are thicker. The right
ventricle (RV) pumps blood into the pulmonary system and the left ventricle (LV) pumps
blood through the aortic valve into the aorta providing the rest of the body with oxygenated
blood. The LV has a thicker wall than the right ventricle, since the RV only has to serve the
pulmonary system and the LV needs to work against a higher pressure to pump blood into
the systemic circulation.
The heart tissue, the myocardium, is composed out of muscle cells, the myocytes. The muscle
cells have the ability to contract to fulfill the pumping function of the heart. To initiate
contraction the myocytes are electrically stimulated. The myocytes are mechanically and
electrically connected through intercalated discs. The overall alignment of the myocytes is
anisotropic throughout the cardiac tissue. This anisotropic alignment gives rise to anisotropy
in both contraction and conduction of the electrical stimulus.
2.2
Electrophysiology of the heart
In a normal muscle cell at rest an electric potential difference is present over the membrane.
This potential difference is called the transmembrane potential and is given by:
Vm = φ i − φ e .
(2.1)
φi represents the intracellular potential, φ e the extracellular potential. At rest the transmembrane potential remains approximately -85mV. Excitation of cardiac tissue occurs when
5
2.2 Electrophysiology of the heart
Theoretical Background
Figure 2.1: Anatomy of the heart, which consists of four chambers: the right and left atrium
and the right and left ventricle.
the transmembrane potential of the myocytes reaches a certain threshold value caused by a
stimulus. A typical myocardial action potential can be seen in figure 2.2. In phase 0 of the
action potential, the transmembrane potential rises to a positive value due to a rapid inflow
of sodium ions. Then, caused by an outward potassium current, in phase 1 a fast decrease in
the membrane potential occurs. The plateau phase of the action potential, phase 2, results
from a slow influx of calcium ions. The outflow of potassium ions finally brings back the membrane potential to its resting value. The amplitude of the action potential in cardiac tissue is
in the order of 100mV. The shape of the action potential differs locally in the myocardium.
Especially the initial repolarization in phase 1 is known to vary in different regions of the
heart. The depolarization will pass on to the neighbour cells through the gap junctions that
connect cardiac cells. In this way the action potential travels over the myocardium.
The electrical activity of the cardiac muscle starts in the sinoatrial (SA) node. The SA node
consists of the so-called pacemaker cells, that have the ability to generate a stimulus. From
the SA node an electric stimulus is generated at a rate of 60 to 100 per minute and conducted
over the right and left atria to the atrioventricular (AV) node. From the AV node the electric
pulse is conducted to the ventricles through the bundle of His. The bundle separates into
two bundles along both sides of the septum, the left and right bundle branches. Both bundle
branches lead to the Purkinje fibers that merge into the ventricular endocardial walls. From
there, the propagation of the depolarization continues over the ventricular walls from cell to
cell at a much slower rate (0.5-1 m/s) than through the conduction system described above
The wave form of the cardiac action potential is not constant. The shape of the action
potential varies in different regions of the heart tissue. The cells in the sinoatrial and atrioventricular node differ from the cells in Purkinje fibers and also differ from the myocardial
cells. This causes changes in the shape of the action potential when traveling from sinoatrial
6
2.2 Electrophysiology of the heart
Theoretical Background
4 0 1
3
4
Vm (mV)
2
Time (ms)
Figure 2.2: Typical form of a myocardial action potential. The dashed lines divide the action
potential in four phases. In phase 0 the cell depolarizes rapidly. Then, in phase 1 a quick
initial repolarization occurs. Phase 2 is the plateau phase of the action potential and in phase
3 the cell repolarizes to the resting state in phase 4. (adapted from [1])
node throughout the myocardium. Also in the myocardial tissue differences in the shape of
the action potential can be found. The duration of the action potential varies throughout the
heart. In figure 2.3 action potential dynamics for different regions in the heart are shown.
Figure 2.3: The shape of the cardiac action potential in different regions of the heart. From
[19]
7
2.3 The electrocardiogram
2.3
Theoretical Background
The electrocardiogram
As a consequence of the depolarization and repolarization of the cardiac muscle cells, potential
differences can be measured with electrodes placed on the skin. Potential differences are
registered for a period of time in a collection of recording electrodes. The signals measured
by these electrodes are collected in an electrocardiogram (ECG).
As a potential difference is measured, at least two electrodes are needed to measure the
ECG. A recording electrode measures the potential difference with a reference electrode.
The connection between the reference and the recording electrode is called a lead. If the
depolarization wave is moving towards the recording electrode, the lead will record a positive
signal. When the recording electrode is placed at a point where the depolarization waves
departs, the signal will be negative. If the lead is perpendicular to the direction of the
traveling depolarization, the lead will give a biphasic signal or no signal at all. The amplitude
of the recorded signal is dependent on the thickness of the muscle.
2.3.1
The 12 lead ECG
The first clinical important ECG measurement setup was developed by Willem Einthoven
[30], called Einthoven’s triangle. The equilateral triangle consists of three limb electrodes,
one on each arm and one electrode on the left leg. The heart can be seen as an electrical
source in the center of this triangle. The leads are assigned I, II and III, and are bipolar
since a potential difference can be calculated between two electrodes. Wilson developed a
way to use unipolar leads for the electrocardiogram. Normally these are determined with
respect to a reference in infinity. Wilson and his colleagues had the idea to use a mean value
of the three limb electrodes as a reference for unipolar leads on the human torso [11, 12].
With Wilsons central terminal as a reference three unipolar leads can be formed with each
limb electrode, the three additional limb leads. Golberger later introduced the augmented
limb leads, the aVR , aVL and aVF where not Wilsons central terminal is the reference, but
the mean of the remaining limb potentials. To measure potentials more closely to the heart,
Wilson introduced six so called precordial leads [13], V 1 to V6 . These are measured with
respect to the central terminal and are placed on the left side of the chest. In the clinic the
12 lead ECG is most commonly used. The 12 lead system consists out of the three standard
leads I II and III, the unipolar leads aV R , aVL , aVF and also contains the six precordial leads.
All the electrodes with lead connections are shown in figure 2.4.
2.3.2
The ECG signal
The standard ECG signal consists of six peak signals each defined with a different letter,
the P, Q, R, S, T and U peaks (see figure 2.5). The P peak results from the depolarization
of the atria. The P-R interval is the time between the depolarization of the atria and the
depolarization of the ventricles. The QRS-complex results from the depolarization of the
ventricles. The T wave displays the repolarization of the ventricles, and the U wave is usually
not present or not important resulting from a rest potential. The origin of the U wave is not
clear but it probably represents ”afterdepolarizations” in the ventricles [19].
8
2.3 The electrocardiogram
Theoretical Background
Figure 2.4: 12 lead ECG electrode placement (from Mybiomedical)
R
T
P
Q
S
Figure 2.5: An example of a typical ECG recording
9
2.3 The electrocardiogram
Theoretical Background
10
Chapter 3
Modeling the electrical activity of
the heart through the torso
In order to model the forward problem of electrocardiography a representation of the actual
cardiac source is used to calculate the body surface potentials. The representations of the
cardiac tissue differ from very detailed microscopic descriptions to the more phenomenological
descriptions, where the active tissue is not described at cellular level but is represented by
a dipole or a collection of dipoles. Modeling the cardiac tissue as a collection of individual
active cells is done by many research groups [9, 17, 5]. The first model to describe the activity
of a single cell was made by Hodgkin and Huxley [2]. The model uses three ionic currents to
describe the flow of ions over the cell membrane. The currents are regulated by gating variables which makes them flow voltage and time-dependent. Later, other even more detailed
models were developed [16, 7].
In the forward problem of cardiac electrophysiology, changes in the ECG to be calculated are
rather determined by the action of a group of cells than by the action of a single cell. Furthermore to model the complete heart built out of single cells could give problems in computation
time and memory. A model that is able to describe the complete cardiac excitation, is the
bidomain model. The bidomain model does not make use of single cells but uses a so called
continuum cell model to describe the properties of the intra- and extracellular space of the
myocardium. The intra- and extracellular spaces are coupled through a membrane where a
transmembrane potential is present. The description of the electrical activation of a single
cardiac cell is used for the continuum cell.
3.1
The bidomain model
The modeling of the forward problem starts with defining a geometry. A schematic drawing
of the parts incorporated in the geometry is shown in figure 3.1 If the assumption is made
that there is no accumulation of charge in a volume, the current flux across the boundary
must be equal to the current produced by the source. Therefore, in general, for every region
the conservation of current is described as:
~ · J~ = Iv
∇
11
(3.1)
3.1 The bidomain model
Modeling electrical activity
∂T
H
∂H
σe ,φ e
σi ,φ i
T
σo ,φo
Figure 3.1: Schematic overview of the regions considered in the forward problem, using a
bidomain approach. H is the cardiac volume and T the torso. ∂H is the cardiac boundary,
∂T the torso surface, σ i,e the intra- and extracellular conductivity tensors respectively and
φi,e are the intra- and extracellular potentials. For the torso T the conductivity is given by
σ o and the potential by φo .
where J~ equals the current density and Iv a produced volume current. From Ohm’s law a
~
constitutive equation can be derived for the current density J.
~
J~ = −σ · ∇φ
(3.2)
with σ the conductivity and φ the potentials of the volume considered.
3.1.1
The heart
The bidomain equations couple two domains in the cardiac volume, the intracellular and
extracellular domains, as a continuous medium, connected by a cellular membrane. The
intracellular domain has a conductivity given by σ i and a potential field given by φi . For the
extracellular domain these properties are indicated by σ e and φe . The conductivity tensors
contain conductivity values in fiber direction and in cross fiber direction, allowing anisotropic
conductivity properties of the myocardium. During an action potential there is an exchange
of ions between the intra- and extracellular space. This transmembrane current is the current
source Iv in excitable media. The derivation of the equations governing the bidomain model
starts with the conservation of current for both domains. From 3.1 for the intracellular domain
the conservation of current gives:
~ · J~i = −Iv
∇
(3.3)
Where Iv is the membrane current per unit volume and J~i the intracellular current density.
This equation describes that the change in current in the intracellular region is equal to the
current Iv leaving the volume. The minus sign indicates that a positive current is defined to
leave the intracellular domain and to flow into the extracellular domain. For the extracellular
region a similar equation can be derived.
12
3.1 The bidomain model
Modeling electrical activity
~ · J~e = Iv
∇
(3.4)
Combining the expressions for the continuity of current for both domains gives:
~ · J~i + ∇
~ · J~e = 0
∇
(3.5)
For the two domains in the cardiac region, the constitutive equation 3.2 is given by:
~ e,i .
J~e,i = −σ · ∇φ
(3.6)
Equation 3.5 is often given in terms of the transmembrane potential V m which is defined as:
Vm = φ i − φ e
(3.7)
Using equation 3.7 and equation 3.6, equation 3.5 can be rewritten to the first equation of
the bidomain model
~ · ((σ i + σ e ) · ∇φ
~ e ) = −∇
~ · (σ i · ∇V
~ m ).
∇
(3.8)
The volume current Iv can be specified by
Iv = A m Im
(3.9)
where Am is the surface to volume ratio of the cell membrane, and I m is the transmembrane
current density per unit area defined as:
Im = C m
∂Vm
+ Iion .
∂t
(3.10)
Where Cm represents the membrane capacitance per unit area and I ion is the sum of the
membrane currents of different ions and is defined per unit area of membrane.
Combining equations 3.3, 3.6, 3.9 and 3.10 the second equation for the bidomain model can
be derived:
~ · (σ i · ∇φ
~ i ) = Am (Cm ∂Vm + Iion )
∇
(3.11)
∂t
This equation can also be written with respect to the transmembrane potential V m . As the
transmembrane potential is related to the transmembrane current, the source term, it is more
convenient to express the equation in terms of V m . This gives the final representation for the
second equation of the bidomain model.
~ · (σ i · ∇V
~ m) + ∇
~ · (σ i · ∇φ
~ e ) = Am (Cm ∂Vm + Iion )
∇
∂t
(3.12)
The ion membrane currents Iion depend on gating variables. These are collected in a column
s and their relation is described by first order differential equations.
˜
∂s
(3.13)
˜ = F (t, s, Vm ) in H
∂t
˜
13
3.1 The bidomain model
3.1.2
Modeling electrical activity
The torso
For the torso the governing equations can be derived in a similar way as for the intra- and
extracellular domains. Conservation of current gives
~ · J~o = Iv
∇
(3.14)
In passive tissue, where no sources are present, the equation above can be simplified using
expression 3.2 derived from Ohm’s law. This results into
~ · σ o · ∇φ
~ o=0
∇
(3.15)
This equation, known as the volume conductor equation or Laplace’s equation, is the general
equation to describe the potential distribution inside a passive medium.
3.1.3
Summary of the bidomain approach
From cell to body surface the forward problem can be solved using the bidomain model.
Summarized the model consists of the following equations:
Vm = φ i − φ e
(3.16)
~ · ((σ i + σ e ) · ∇φ
~ e ) = −∇
~ · (σ i · ∇V
~ m)
∇
(3.17)
~ · (σ i ∇V
~ m) + ∇
~ · (σ i · ∇φ
~ e ) = Am (Cm ∂Vm + Iion )
∇
∂t
(3.18)
The gating variables, regulating the ion membrane currents I ion , are collected in the column
s. They are described by first order differential equations.
˜
∂s
in H
(3.19)
˜ = F (t, s, Vm )
∂t
˜
The potential distribution inside the torso is described with Laplace’s equation.
~ · σ o · ∇φ
~ o=0
∇
(3.20)
The boundary conditions needed to solve the set of equation are listed below. From the
intracellular region no current flows into the torso.
~ i ) · ~n = 0
(σ i · ∇φ
on
∂H
(3.21)
Between the extracellular and torso region a current balance is kept through the following
boundary condition
~ e ) · ~n = −(σ o · ∇φ
~ 0) · n
(σ e · ∇φ
on
∂H.
(3.22)
Because no potential drop is possible across a boundary the potentials of the extracellular
domain and the torso have to be equal at the boundary.
14
3.2 Decoupling the problem
Modeling electrical activity
φe = φ o
on ∂H
(3.23)
Considering the torso to be surrounded by air, with conductivity zero, no current leaves the
torso through the surface.
(σ o · ∇φo ) · nT = 0 on ∂T
(3.24)
3.2
Decoupling the problem
Within the cardiac region, the bidomain equations are generally used to calculate the propagation of action potentials. The bidomain model describes the membrane potential as a
function of the location ~x and of time t. A typical cardiac action potential at a certain ~x = ~x 0
is given in figure 3.2.
potential (mV)
10
0
-90
0
250
time (ms)
Figure 3.2: A typical shape of the membrane potential V m (~x0 , t) described by the bidomain
model
Because of the fact that the upstroke of the action potential is so steep it will only cover a
few cells at the same moment. Therefore, to describe the propagation of the action potential
realistically, a spatial grid size in the order of a cell is needed. Furthermore the action potential
has a duration of approximately 250 ms and the upstroke has a duration in the order of a
millisecond. To be able to closely describe the dynamics of the action potential a timescale
in the order of a microsecond is needed. In conclusion, the problem is computationally very
expensive, both in time and in spatial domain. Therefore the aim is to find a less time and
memory consuming method. Several research groups have tried to achieve a faster and more
simplified way to the solution by decoupling the problem [31, 22]. In these works the model
of activation of the myocardium is divided into two steps. The first step is to determine the
activation sequence in the heart. The second step involves the modeling of the action potential
dynamics. For the first step for example the moment of depolarization can be used to simplify
the activation model. The activation sequence can be determined in several ways, Miller used
an estimation from literature based on measurements by Durrer et al. [31, 10], Van Dam uses
a shortest path algorithm [24]. Another method is to use the Eikonal equation, derived from
the bidomain model by Colli-Franzone et al. [23]. This equation describes the motion of the
depolarization front through the myocardium. The Eikonal equation describes the moment of
the upstroke of the action potential as a moving front without using a detailed description of
15
3.2 Decoupling the problem
Modeling electrical activity
the ionic current. It also provides the ability to incorporate anisotropic conduction properties
of the cardiac tissue.
3.2.1
Eikonal equation
The original description of the Eikonal equation in Franzone et al. [23] has been rewritten by
Kerckhoffs [25] by assuming the ratio between conductivities in longitudinal and transverse
direction to be equal for intra- and extracellular media. The anisotropy ratio m is given by
σti,e
=m
σli,e
with subscript t denoting the transverse direction, and subscript
direction. The conductivity tensors can be written as:
(3.25)
l
the longitudinal or fiber
σ i = σli (~el~el + m(~et1~et1 + ~et2~et2 )) = σli M
(3.26)
σ e = σle M
(3.27)
with σ i the intracellular conductivity tensor and σ e the extracellular conductivity tensor.
Vector ~el is a unit vector in fiber direction and ~e t1 and ~et2 are unit vectors in transverse
direction.
These assumptions result in the following equation that has to be solved for t dep :
cf
q
~ dep · M · ∇t
~ dep − k0 ∇
~ · (M · ∇t
~ dep ) = 1
∇t
(3.28)
where cf represents the velocity of the depolarization wave along myofiber direction, k 0 is a
constant that determines the influence of wave front curvature on the wave velocity.
Boundary conditions used to solve the equation are
tdep = 0
at
Γstim
(3.29)
and
~n · M · p~ = 0
on
Γext
(3.30)
where Γstim is the site where the depolarization is initiated, the stimulation site, and Γ ext is
the boundary of the domain assumed that it is electrically insulated.
The advantages of the Eikonal model over the bidomain approach are the reduction in calculation time, the only parameters are the activation times of the cardiac fibers, and that
local velocity differences can be modeled. However, the advantages of the Eikonal model also
bring some disadvantages. The model only describes the moment of the upstroke of the action
potential, this means that the repolarization is not included. To be able to describe the complete action potential dynamics an additional description of the action potential dynamics
has to be coupled to the activation times. Because the shape of the action potential is not
constant through the cardiac tissue it has to be estimated. Furthermore the model is a more
phenomenological description and is more difficult to relate to the physiological basis of the
problem.
16
3.2 Decoupling the problem
3.2.2
Modeling electrical activity
Common methods
When the activation sequence on the myocardium is known there are different ways to model
the forward problem. The conductive regions incorporated in the model can either be described by volumes or by surfaces. If volumes are considered, a volume method is used. If
only interfaces between different regions are comprised, a surface method is applied. Here a
short overview of common methods to solve the forward problem is given. A more complete
overview can be found in [27] and [15].
Volume method
Volume methods can be subdivided into two approaches. The cardiac activity can be represented by either equivalent current sources, or by potentials prescribed on the cardiac surface.
Volume methods considering current sources are based on solving the governing equation for
the cardiac source
~ · σ e · ∇φ
~ e = −Iv .
∇
in H
(3.31)
And for the torso, where no sources are present the following equation is used
~ · σ o · ∇φ
~ o = 0.
∇
in
T
(3.32)
With the volume method the activation can be defined through the complete heart. This can
be done by defining the volume current given in equation 3.31 and switching it on according
to the predefined activation sequence through the myocardium.
Solving the complete volume model
To calculate the activation of the heart through the torso to the body surface, the finite
element method (FEM) can be used. The governing equations to solve the problem using the
FEM are given below.
For the complete volume, H and T , equation 3.31 is solved. The weak form of this equation
is given by
Z
~ · σ · ∇φ(~
~ x, t)dV = −
w(~x)∇
H
T
T
Z
H
w(~x)Iv (~x, t)dV
T
(3.33)
T
with w(~x) a weighting function, the potential φ is φ e in H and φo in T. The conductivity
tensor σ is σe in H and σ o in T. For convenience the dependence on ~x and t is left out in the
remainder of this section. When applying Greens theorem to this, the following equation is
derived:
Z
∂T
~ · ~ndS −
wσ ∇φ
Z
H
~ · (∇w)dV
~
(σ e ∇φ)
=−
T
T
Z
wIv dV
H
T
(3.34)
T
For the torso T there is no source term, so the integral on the right side only holds for the
heart H. Together with a zero flux boundary condition at ∂T , which cancels out the integral
on the left, the equation becomes
17
3.2 Decoupling the problem
Modeling electrical activity
Z
~ · (∇w)dV
~
(σ ∇φ)
=
H
T
Z
wIv dV
(3.35)
H
T
To make the problem well-posed the potential is prescribed at one point on the outer boundary
∂T
φ(xref ) = 0
for xref
at
∂T
(3.36)
Equations 3.35 the basis for solving the forward problem using the FEM.
Volume method for thorax only
When using the finite element method, the torso volume is filled with volume elements. In this
situation, the cardiac volume is left out of the solution, only the space between the epicardium
and the torso surface is considered. Equation 3.31 disappears in this case, because in the torso
no sources are present. Equation 3.32 needs to be solved.
The derivation of the equations is similar as in the volume method. The weak form becomes
Z
Z
~ · (∇w)dV
~
~ · ~ndS − (σ o · ∇φ)
=0
(3.37)
wσ o · ∇φ
T
∂T
The integral on the left disappears because of a zero flux boundary condition at ∂T , resulting
in
Z
~ · (∇w)dV
~
(σ o · ∇φ)
=0
(3.38)
T
For the heart surface ∂H the following boundary condition is given
φe = φe,0
on
∂H
(3.39)
Surface method
If the forward problem is solved using a surface approach only the boundary of the heart, ∂H,
is taken into account. The activation of the heart is represented on the cardiac surface. To
solve the surface method, a boundary element method (BEM) is used to solve the problem
numerically.
Solving the surface model
When the BEM is used, the torso volume T is not described, only the heart and torso surface
∂H and ∂T are numerically represented. In surface methods isotropic conduction properties
are assumed for the volume of the cardiac wall and the torso. This means that the tensor σ
becomes a scalar σ for every domain. The BEM reduces the amount of nodes that is needed
to calculate the solution by expressing the problem as function of the boundary instead of
describing it in space using Laplace’s equation. By assuming isotropic conduction properties
and homogeneity of the thorax, the governing equation for potential distribution due to the
cardiac electrical activity can be expressed in terms of surface integrals only. A representation
18
3.2 Decoupling the problem
Modeling electrical activity
of the source Iv is derived on the heart surface ∂H, also called an equivalent surface source.
For a detailed derivation of the governing equations describing the surface method, see [21].
Since all equations are expressed in terms of surface integrals, they can be combined into a
single equation. The equation contains a function A, the transfer function which transforms
the intracellular potential φi originating from the sources on the cardiac wall, into potentials
on the boundary ∂T . This function contains information on the conductivity values of the
different parts of the volume conductor model and of the geometries of the compartments.
φ(~x0 , t) =
Z
A(~x, ~x0 , t)φi (~x, t)dS
S
with x~0 a point on the torso boundary ∂T .
19
(3.40)
3.2 Decoupling the problem
Modeling electrical activity
20
Chapter 4
Two dimensional test problems
In chapter 3, the volume method to calculate the forward problem was described. With the
volume method the potentials inside a volume conductor caused by a volume current source
are calculated. Here, the volume method will be analyzed and results will be compared with
the results of simulations where the source is represented by surface potentials. The equations
governing both solution methods are given in section 3.2.2. To gain insight in the functioning
of the volume method, a few test problems are defined. The domain that is used in these
test problems, a square, is a simple representation of a thorax. Various source descriptions
are considered, which represent cardiac activity. Potentials distributed from these sources
are considered at the right boundary of the domain, representing the body surface. At first,
the field of a simple dipole source is calculated analytically and a numerical approximation
is made. In a second test problem the source is extended to a dipole layer. And a third test
problem contains the comparison of solutions of the volume method and the surface method.
4.1
4.1.1
Test 1 (dipole)
Analytical description
A combination of two point sources with opposite current strength I 0 and −I0 is called a
dipole. In the previous chapter it was explained that dipoles are often used as a source model in
electrocardiographic modeling. The analytical solution for the potential produced by a dipole
in a 2D domain starts with the description for a single point source. The potential generated
by a point source has spherical symmetry and is usually derived in a three dimensional domain.
However since the concept of this chapter is to give more insight, here, a two dimensional
description is given. The current density caused by a monopole with current strength I 0 in a
two dimensional space is given by the current passing the surface.
I0
~er
J~ =
2πr
(4.1)
with ~er a unit vector in the radial direction. In chapter 3 the following constitutive equation
was assumed using Ohm’s law.
J~ = −σ∇φ
(4.2)
21
4.1 Dipole
Two dimensional test problems
Because of the fact that the problem is rotationally symmetric only the radial component is
non-zero, this leads to
dφ
I0
=
(4.3)
dr
2πr
After integration with respect to r and defining the potential at infinity as zero, it follows for
the potential generated by a monopole in 2D
−σ
I0 ln(r)
(4.4)
2πσ
For a dipole consisting of two opposite point charges separated by a distance 2d the potential
in a point P (x, y) is found by superposition of the potentials of the two point sources. In
figure 4.1 the configuration of the dipole is drawn schematically.
φ=
y
+ I0
R1
x
r
2d
- I0
R2
P(x,y)
Figure 4.1: The position of a dipole consisting of two opposite charges I 0 and −I0 separated
by a distance 2d
Using equation 4.4 for the potential for a single point source, the superposition of the two
opposite sources in figure 4.1 gives
φ(x, y) =
I0
R1
ln( )
2πσ
R2
(4.5)
For a line x = x0 the potential is given by
p
(y − d)2 + (x0 )2
I0
φ(x0 , y) =
ln p
2π
(y + d)2 + (x0 )2
(4.6)
The result of equation 4.6 can be seen in figure 4.3.
4.1.2
Numerical Implementation
A finite element approach is made using SEPRAN, and compared to the analytical solution
described above. A dipole was simulated by assigning opposite current strengths I 0 and −I0
to two points in a homogeneous mesh. When using finite elements the value of the current
strength is interpolated over the elements connected to the active node.
22
4.1 Dipole
Two dimensional test problems
In chapter 3 the weak form for active region in the volume method was derived from Poisson’s
equation for active tissue.
Z
~ · ~ndS −
wσ ∇φ
∂T
Z
~ · (∇w)dV
~
(σ e ∇φ)
=−
H∩T
Z
wIv dV
(4.7)
H∩T
The right integral of this equation describes the source. The current that was prescribed at
the active node is interpolated with the use of linear shape functions. Within each element
an approximation of the current strength I v is given by
Ive =
4
X
Ni (x, y)Ii
(4.8)
i=1
With Ni the shape functions of each element node in a global coordinate system. Generally
a local coordinate system is used, with coordinates −1 < ξ < 1 and −1 < η < 1. The shape
functions are for instance given by
N1 =
1
(1 − ξ)(1 − η)
4
(4.9)
Since only one node in the element is given a value I v0 for Iv the following can be derived
Ive (ξ, η) = N1 (ξ, η)Iv0 =
I v0
(1 − ξ − η + ηξ)
4
(4.10)
The total source strength I0 of one element can be calculated using the expression for I ve and
is given by
Z
Ive (x, y)dΩ
(4.11)
I0 (x, y) =
Ωe
Filling in the terms and writing ξ = x/L 1 and η = y/L2 , with L1 and L2 the element
dimensions of Ωe in x and y direction, gives the following equation
I0 (x, y) =
Z
Ωe
x
y
xy
I v0
(1 −
−
+
)dΩe
4
L1 L2 L1 L2
(4.12)
This equation has the following solution
I v0
L1 L2
(4.13)
4
In this test problem a two dimensional mesh is used. Because quadrilateral elements are used
with a bilinear interpolation, the current given to the node is integrated over 4 elements with
equal side lengths Le . This gives a surface source of
I0 =
I0 = Iv0 L2e
(4.14)
with L2e the surface of one element. This means that when the element size changes, the dipole
strength changes. For illustration, two simulations are performed with different element sizes.
A total current strength of 4 · 10−4 [A · m−2 ] is used in both simulations. Simulation 1 is
performed with elements with sides L e = 0.02 m and in simulation 2 the same domain is used
23
4.1 Dipole
Two dimensional test problems
but elements with sides Le = 0.01 m are used. Using equation 4.14 the current strength I 0
is 1 A in simulation 1 and 4 A in simulation 2. Poisson’s equation is solved to derive the
potential throughout the domain. A zero flux boundary condition is applied at the outer
boundary ∂T .
In a third simulation the original domain is made two times larger. The settings used for the
three simulations are displayed in table 4.2.
Simulation
1
2
3
element size[m]
0.02
0.01
0.02
domain
2x2
2x2
4x4
I 0 [A]
1
4
1
Iv0 [A · m−2 ]
4e−4
4e−4
4e−4
Figure 4.2: Settings for the simulations in SEPRAN
4.1.3
Results
The results of the three simulations can be seen in figure 4.3.
−5
1.5
x 10
Potentials generated by a current dipole
1
potential (V)
0.5
0
−0.5
−1
−1.5
−1
−0.5
0
y coordinate
0.5
1
Figure 4.3: Potentials generated by a dipole derived with three simulations and the analytical
solution. Simulation 1 is given with the dashed and dotted line (− · − · −) , simulation 2
with the dashed line (− − − −), simulation 3 with the dotted line (· · ·) and the analytical
solution is given with the solid line ( ).
In the center of the figure all the results look similar, however, at the boundaries of the domain
differences in the potential are visible. The solutions from simulation 1 and 2 are similar, the
result of simulation 3 approaches the analytical solution. Note that the domain of simulation
3 is in fact larger than displayed in figure 4.3.
4.1.4
Discussion
As can be seen in figure 4.3, the results of the simulations differ from the analytical solution
at the boundary of the domain. This is probably a result of the boundary conditions that are
used in the simulations. The zero flux boundary condition forces the gradient of the potential
to be aligned along the boundary. In the analytical solution the potential is assumed to
24
4.2 Dipole layer
Two dimensional test problems
be zero in infinity. Therefore it can be expected that at the boundary of the domain the
solution of the simulations differs from the analytical solution. And when the dimensions of
the domain are increased, the solution is expected to move closer to the analytical solution.
This is shown by the third simulation where a larger domain is used.
4.2
Test 2 (Dipole layer)
A source that is often used to model the activity of tissue, is the dipole layer. The depolarization wave front spreading throughout the myocardium can be represented by a dipole layer
with the vector normal to the curvature of the wave front.
4.2.1
Analytical description
A dipole layer can be seen as a line of dipoles. In this test the dipoles are oriented in the
same direction. The solution of the dipole potential can be used to formulate the potential
generated by dipoles arranged in a line along x. The dipole layer consists of a area A of
length L and width d with positive sources and an area B with negative sources, separated
by a distance 2a. In figure 4.4 the dipole layer is drawn schematically. The potentials are
determined in a line along y located at x = x 0 . The total current strength I per area is
153[A]. The potential generated by a dipole layer at a point (x,y) with distances R1 and R2
from the poles and a current strength I 0 for every pole is given by
φ(x, y) =
1
2πσ
Z Z
y
I0 ln(
x
R1
)dxdy
R2
The integration domain is the surface of the dipole layer.
25
(4.15)
4.2 Dipole layer
4.2.2
Two dimensional test problems
Numerical implementation
To simulate the potential generated by a dipole layer, simulations were performed using
SEPRAN. First a simulation is performed using the volume method described in chapter 3.
To obtain a dipole layer source, nodes in area A are given a current strength I 0 while in area
B the nodes have a current strength of −I 0 . This results in a total current strength equal
to the current strength used in the analytical problem. As was explained in the previous
chapter, for the passive domain Laplace’s equation is solved to obtain the potentials that are
caused by the source. A zero flux boundary condition is used at the outer boundary ∂T .
Furthermore the potential was set to zero at one point on ∂T .
P0
A
d
2a
B
L
Y
X0
X
Figure 4.4: Schematic drawing of the geometry used for the dipole layer. With the dipole
layer consisting of area A and area B which are prescribed opposite current strengths, L = 1
m, a = 0.04 m, d = 0.06 m and the domain has dimensions 2 m by 2 m. In the node in the
top left corner of the domain (P0 ) the potential is set to zero.
4.2.3
Simulations performed
In the first simulation the domain is filled with 200 rectangular shaped elements, and in the
second simulation the same grid size was used but the elements shape is triangular and this
generates an irregular mesh. In figure 4.5 the two different element types are plotted in the
problem domain.
Figure 4.5: The mesh formed by rectangular elements (top) and triangular elements selected
from one half of the dipole layer.
In a third simulation the effect of an unequal source size on the solution is shown. For area
26
4.2 Dipole layer
Two dimensional test problems
A the current size is kept at 1, but for area B a decrease in current strength of 0.5% and 1%
was applied. In this simulation rectangular elements are used.
For the numerical approximation made with a rectangular grid, potentials are determined in
a line along y at x = x0 (see figure 4.4).
4.2.4
Results
In figure 4.6, the results of the potentials determined in a line along y at x = x 0 are shown
for the analytical solution and the numerical approximation. The result of the numerical
approximation is shifted 1.05 · 10−3 V to make it symmetric around zero, and to be able to
compare with the analytical solution.
-3
1.5
x 10
1
potential (V)
0.5
0
-0.5
-1
-1.5
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
y coordinate
0.4
0.6
0.8
1
Figure 4.6: Plot of the potential determined alon line x = x 0 with the solid line representing
the analytical solution and the dashed line the numerical solution.
As can be seen the solutions are similar in the center and vary at the boundary of the domain.
In figure 4.7 surface equipotential plots of the potentials calculated in the first and the second
simulation are shown.
27
4.2 Dipole layer
Two dimensional test problems
LEVELS
-3.2978E-03
-3.0724E-03
-2.8470E-03
-2.6216E-03
-2.3963E-03
-2.1709E-03
-1.9455E-03
-1.7201E-03
LEVELS
LEVELS
-1.4947E-03
-1.2693E-03
Contour levels of solution
Regular mesh
-3.2978E-03
-1.3130E-02
-3.0724E-03
-1.2474E-02
-2.8470E-03
-1.1817E-02
-2.6216E-03
-1.1161E-02
-2.3963E-03
-1.0504E-02
-2.1709E-03
-9.8475E-03
-1.9455E-03
-9.1910E-03
-1.7201E-03
-8.5345E-03
-1.4947E-03
-7.8780E-03
-1.2693E-03
-7.2215E-03
-1.0439E-03
-6.5650E-03
-8.1855E-04
-5.9085E-03
-5.9317E-04
-5.2520E-03
-3.6778E-04
-4.5955E-03
-1.4239E-04
-3.9390E-03
8.2991E-05
-3.2825E-03
3.0838E-04
-2.6260E-03
5.3376E-04
-1.9695E-03
7.5915E-04
-1.3130E-03
9.8453E-04
-6.5650E-04
1.2099E-03
0.0000E+00
scaley:
7.500
scalex:
7.500
time t:
0.000
Contour levels of solution
Irregular mesh
scaley:
7.500
scalex:
7.500
scaley:
0.000
time t:
scalex:
time t:
Contour levels of solution
-1.0439E-03
-8.1855E-04
-5.9317E-04
-3.6778E-04
-1.4239E-04
8.2991E-05
3.0838E-04
5.3376E-04
7.5915E-04
9.8453E-04
1.2099E-03
7.500
7.500
0.000
Figure 4.7: Contour plot of potentials calculated using the volume method with a regular
mesh (left) and an irregular mesh (right)
LEVELS
-3.0000E-03
-2.5650E-03
-2.1300E-03
-1.6950E-03
-1.2600E-03
-8.2500E-04
-3.9000E-04
4.5000E-05
LEVELS
LEVELS
LEVELS
4.8000E-04
9.1500E-04
Contour levels of solution
-3.0000E-03
-3.0000E-03
-3.0000E-03
-2.5650E-03
-2.5650E-03
-2.5650E-03
-2.1300E-03
-2.1300E-03
-2.1300E-03
-1.6950E-03
-1.6950E-03
-1.6950E-03
-1.2600E-03
-1.2600E-03
-1.2600E-03
-8.2500E-04
-8.2500E-04
-8.2500E-04
-3.9000E-04
-3.9000E-04
-3.9000E-04
4.5000E-05
4.5000E-05
4.5000E-05
4.8000E-04
4.8000E-04
4.8000E-04
9.1500E-04
9.1500E-04
9.1500E-04
1.3500E-03
1.3500E-03
1.3500E-03
1.7850E-03
1.7850E-03
1.7850E-03
2.2200E-03
2.2200E-03
2.2200E-03
2.6550E-03
2.6550E-03
2.6550E-03
3.0900E-03
3.0900E-03
3.0900E-03
3.5250E-03
3.5250E-03
3.5250E-03
3.9600E-03
3.9600E-03
3.9600E-03
4.3950E-03
4.3950E-03
4.3950E-03
4.8300E-03
4.8300E-03
4.8300E-03
5.2650E-03
5.2650E-03
5.2650E-03
5.7000E-03
5.7000E-03
5.7000E-03
scaley:
7.500
scalex:
7.500
time t:
0.000
Contour levels of solution
scaley:
7.500
scaley:
7.500
scalex:
7.500
scalex:
7.500
time t:
0.000
Contour levels of solution
scaley:
0.000
time t:
scalex:
time t:
Contour levels of solution
1.3500E-03
1.7850E-03
2.2200E-03
2.6550E-03
3.0900E-03
3.5250E-03
3.9600E-03
4.3950E-03
4.8300E-03
5.2650E-03
5.7000E-03
7.500
7.500
0.000
Figure 4.8: Contour plot of potentials generated by a dipole layer with left: I 0 = −I0 , middle:
0.5% decrease in area B and right: 1% decrease in current strength I 0 in area B.
What can be noticed from figure 4.7 is that when a regular mesh is used the potential field is
symmetric, and when the elements are irregularly shaped, the potential field is asymmetric.
The solution seems to be out of balance in the second simulation in figure 4.7.
The result of experiment three can be seen in figure 4.8. The picture on the left in this figure
is the same as the left picture in figure 4.7, however the scale is altered to be able to compare
the solution with the other results in the figure. From figure 4.8 it can be noticed that when
the source strength is unequal for the two layers, the potential distribution is asymmetric.
Vector plots were made from the top left quarter of the domain, these can be seen in figure
4.9. In figure 4.9 there is a flux visible through the point in the upper left corner where the
potential is prescribed.
28
4.2 Dipole layer
Two dimensional test problems
factor:
4.000
factor:
scaley:
15.000
scaley:
scalex:
15.000
scalex:
15.000
time t:
0.000
time t:
0.000
Vector plot of gradient
Vector plot of gradient
factor:
scaley:
Vector plot of gradient
4.000
15.000
scalex:
15.000
time t:
0.000
Figure 4.9: Vector plots made from the gradient in potential in the top left quarter of the
domain. Top left :I0 = −I0 , top right: 0.5% decrease in current in area B, bottom left: 1%
decrease in current strength I0 in area B.
29
4.000
15.000
4.3 Volume source versus surface source
4.2.5
Two dimensional test problems
Discussion
In figure 4.6 the analytical solution is compared with a numerical approximation. The results
are similar in the center but vary at the boundary of the domain. This can be expected
because of the zero flux boundary condition applied in the numerical problem.
In figure 4.7 and 4.8 the results of test 2 can be seen. In figure 4.8 the positive and negative
part of the dipole layer are made unequal in current strength. As can be seen, already a small
deviation gives a change in the potential distribution. When the source strength is out of
balance there will be flow possible out of the domain through the top left corner of the outer
boundary where the potential is set to zero. The asymmetric potential distribution seen in
figure 4.7 in case of an irregular mesh can therefore be declared by the fact that the irregular
mesh creates an imbalance in current strength between the layers.
4.3
Test 3 (Volume source versus surface source)
To compare the solution obtained using a volume source with the solution obtained using a
surface source, the solution of simulation 1, in part Test 2, is used as input for the present
simulation. A mesh was created with the same dimensions, and a circular saving in the center. This hole has a radius of 0.5m and represents the outer surface of the heart. In Matlab
a value for the potential for every node on the circular boundary is determined from the
solution in the closest node in the volume method. These values for the potential are used
as boundary condition on the circular boundary, the cardiac surface ∂T . A solution for the
potential was determined solving Laplace’s equation. In an additional simulation the hole is
filled with elements, representing a heart of which the volume is filled. On the boundary of
the hole potentials are prescribed, and the area inside the hole is conductive.
4.3.1
Results
Contour plots of the potentials in the three simulations are shown in figure 4.10.
Potentials along the right boundary of the domain are plotted for both methods in figure
4.11, to be able to compare the two solutions more directly.
4.3.2
Discussion
In the third test situation the volume method is compared with the surface method. From
figure 4.9 it can be seen that the approaches show similar potential patterns throughout the
domain. Also when the area inside the hole is filled with elements, the pattern looks similar.
Inside the hole the potentials show differences, this can be explained by the fact that that
area is modeled as a passive conducting region. However, in the forward problem, the interest
lies in the potential distribution from the heart surface to the torso surface.
30
4.3 Volume source versus surface source
Two dimensional test problems
LEVELS
-3.2978E-03
-3.0724E-03
-2.8470E-03
-2.6216E-03
-2.3963E-03
-2.1709E-03
-1.9455E-03
-1.7201E-03
LEVELS
LEVELS
LEVELS
-1.4947E-03
-1.2693E-03
Volume source
Contour levels of solution
-3.2978E-03
-3.2978E-03
-3.2978E-03
-3.0724E-03
-3.0724E-03
-3.0724E-03
-2.8470E-03
-2.8470E-03
-2.8470E-03
-2.6216E-03
-2.6216E-03
-2.6216E-03
-2.3963E-03
-2.3963E-03
-2.3963E-03
-2.1709E-03
-2.1709E-03
-2.1709E-03
-1.9455E-03
-1.9455E-03
-1.9455E-03
-1.7201E-03
-1.7201E-03
-1.7201E-03
-1.4947E-03
-1.4947E-03
-1.4947E-03
-1.2693E-03
-1.2693E-03
-1.2693E-03
-1.0439E-03
-1.0440E-03
-1.0440E-03
-8.1855E-04
-8.1857E-04
-8.1857E-04
-5.9317E-04
-5.9318E-04
-5.9318E-04
-3.6778E-04
-3.6780E-04
-3.6780E-04
-1.4239E-04
-1.4241E-04
-1.4241E-04
8.2991E-05
8.2975E-05
8.2975E-05
3.0838E-04
3.0836E-04
3.0836E-04
5.3376E-04
5.3374E-04
5.3374E-04
7.5915E-04
7.5913E-04
7.5913E-04
9.8453E-04
9.8452E-04
9.8452E-04
1.2099E-03
1.2099E-03
1.2099E-03
scaley:
7.500
scalex:
7.500
time t:
0.000
Contour levels of solution
Surface source
scaley:
7.500
scalex:
7.500
time t:
0.000
Contour levels of solution
scaley:
7.500
scalex:
7.500
Surface source, filled
scaley:
0.000
time t:
scalex:
time t:
Contour levels of solution
-1.0439E-03
-8.1855E-04
-5.9317E-04
-3.6778E-04
-1.4239E-04
8.2991E-05
3.0838E-04
5.3376E-04
7.5915E-04
9.8453E-04
1.2099E-03
7.500
7.500
0.000
Figure 4.10: Contour plot of potentials calculated using the volume method (left), the surface
method (middle) and the surface method with the cavity filled with elements (right)
-3
1
x 10
0
potential
-1
-2
-3
Volume source
Surface source
-4
Surface source filled
-5
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
y coordinate
0.4
0.6
0.8
1
Figure 4.11: Absolute potentials along the right boundary of the domain for the volume
source (dashed and dotted line), the surface source (solid line) and the filled surface source
(dashed line)
31
4.4 General Discussion
4.4
Two dimensional test problems
General Discussion
The three tests problems described above were performed to obtain more insight into modeling
the potential generated by current dipole sources inside a volume conductor. From the first
two tests it can be seen that when the source is in balance, the volume source generates a
potential distribution as can be expected from the analytical solutions. However, as soon as
there is an imbalance in the source strength, the potential distribution becomes asymmetric.
In test three it can be seen that the solutions obtained with a surface source and with the
volume source match closely in the thorax domain (T).
The aim is to describe the potentials generated by the activity of a realistically shaped heart
in a realistic torso geometry. Since the solution based on current dipole sources has problems
with an irregular mesh, it was chosen to use a source represented by potentials on the cardiac
wall to model a 3D situation in the following chapter.
32
Chapter 5
From cardiac activation to body
surface potentials
In chapter 4 it was motivated that the volume method with the use of current dipole sources,
is sensitive to imbalances in current sources and sinks due to, for example, irregular meshing.
Prescribing cardiac electrical activity through voltage sources seems more robust. Therefore,
in this chapter potentials are prescribed throughout the cardiac wall.
A mesh of a dog heart described in Kerckhoffs et al. [26] is placed inside a stylized torso.
Activation times are determined in the wall of the heart using the Eikonal equation. The
activation times are coupled to an action potential and by using Laplace’s equation for the
potential distribution, potentials are calculated on the surface of the torso. The results are
compared to simulations performed by Miller et al. [31] who used a multiple dipole approach.
5.1
5.1.1
Methods
Activation of the heart
The geometry of the heart that is used is taken from Kerckhoffs et al. [26]. It contains the
right and left ventricle of a dog heart, the atria are not taken into account. The Eikonal
curvature equation described in chapter 3 is used to calculate the activation times in each
node of the heart mesh. The activation is initiated at four points on the cardiac wall central
at the left septal surface, posterior paraseptal at one third the distance from apex to base,
at the anterior paraseptal wall below the mitral valve and near the right anterior papillary
muscle.
The mesh that is made of this geometry is build out of 33536 (hexahedral) elements. The
activation times that are calculated for the heart mesh are shown in figure 5.1. The total
depolarization of the ventricles lasts approximately 60ms. For comparison, the activation
times measured by Durrer et al. [10] in isolated human hearts are shown in figure 5.2.
33
5.1 Methods
From cardiac activation to body surface potentials
60
50
40
60
50
40
30
30
20
20
10
10
0
0
Figure 5.1: Activation sequence according to Kerckhoffs et al.[25]
Figure 5.2: Isochronic surfaces of cardiac activation, measured by Durrer et al. in isolated
human hearts [10].
5.1.2
Heart inside the torso
The smooth mesh of the heart is placed inside a stylized torso. The torso is constructed
from two eccentric half cylinders, one with a radius of 340 mm and one with a radius of 385
mm with a shifted center to create a boundary with less curvature to represent the back of
the torso. A front and top view of the torso contour containing the heart is shown in figure
5.3. The height of the resulting torso is 500 mm, the width is 300 mm and the depth is
approximately 200 mm.
The measures of the thorax are estimated to be comparable to the measures used by Miller
et al. The mesh that forms the torso is build out of 242190 elements, a tetrahedral element
shape is used. The anatomical position and orientation of the heart inside the torso differs for
every individual, and since in this situation no realistic geometries are used, the right position
and orientation of the heart is difficult to motivate.
To be able to make comparisons with Miller et al. the heart is placed in a similar position
34
5.1 Methods
From cardiac activation to body surface potentials
as described in their work. The coordinate system contains orthogonal vectors ~e x , ~ey , ~ez ,
where ~ez coincides with the LV long axis, ~ex points at the LV free wall. ~ey is orientated
perpendicular to ~ex and ~ez such that ~ey = ~ez × ~ex . The origin was chosen on the LV long axis
at 25 mm below the basal plane. With respect to the torso, the heart is rotated around the z,
y and x axis with angles π/8, π/4 and π/4 respectively. Subsequently the heart is translated
to be placed at a more anatomic location inside the torso. From the center of the bottom
plane of the torso (x,y,z) = (0,0,0) the origin of the heart is placed at z = 300 mm, y = -90
mm and x = 40 mm. The result of the location and orientation of the heart inside the torso
can be seen in figure 5.3.
Front view
Top view
Figure 5.3: Left: front view of the torso boundaries containing the heart. Right: a top view
of the torso containing the heart. The electrode positions of the standard 12 lead system are
indicated with the squared markers. R, L and F are the three limb electrodes and V 1 to V6
the precordial electrodes.
35
5.1 Methods
From cardiac activation to body surface potentials
In the torso, the nodes that lie inside the wall of the heart are assigned a value for the
activation time. Because the torso nodes do not coincide with the heart nodes, the activation
time has to be interpolated for the torso. This is done by looking in a region with a radius 3
mm around every node in the torso and selecting the nodes of the heart mesh that are within
that radius. The value of the activation time for the node in the torso mesh is determined
from a weighted value of the activation times of the surrounding nodes in the heart mesh.
PN
i=1 (1
tdep = PN
ri
rmax )ti
ri
− rmax
)
−
i=1 (1
(5.1)
with tdep the activation time to be calculated for the torso node, N the surrounding nodes
of the heart mesh, ri the distance of node i to the node in the torso mesh, r max the maximum distance of the surrounding nodes and t i the activation time of node i in the heart mesh.
Now only the time of depolarization is known in the nodes describing the cardiac wall, however
to describe the action potential dynamics, an action potential has to be coupled to these
depolarization times. The action potential shape is shown in figure 5.4. The action potential
starts at a resting potential of -90 mV and rises to 10.45 mV to repolarize back to the resting
potential in approximately 260 ms.
Potential (mV)
20
0
-20
-40
-60
-80
-100
0
50
100
150
200
250
300
Time (ms)
Figure 5.4: The shape of the actionpotential coupled to the depolarization times in the
simulations.
5.1.3
Determining the ECG
A simulation is performed where the potentials are prescribed on 1430 nodes in the torso,
representing the myocardium. Potentials are calculated on the torso surface using Laplace’s
equation. A no flux boundary condition is applied to the torso surface. Surface equipotential
maps are made of the front side of the thorax. The voltages are calculated with respect to
Wilsons central terminal which is the mean value of the potential of the three limb electrodes:
φCT =
φR + φ L + φ F
3
(5.2)
The potentials were normalized to obtain a scale from -1 mV to 1 mV, the multiplication factor
36
5.2 Results
From cardiac activation to body surface potentials
is given above each map. During depolarization 8 time steps are shown, during repolarization
4 time steps are considered.
To determine an ECG signal, in the clinic, electrodes are placed on the body surface. To
reproduce ECG signals, potentials were measured at the outer surface of the torso. Nine
surface sites were chosen, in comparison to the sites chosen in Miller et al., to calculate a 12
lead ECG. The surface sites used for the calculation of the ECG are sketched in figure 5.3.
R, L and F represent the limb leads and V 1 to V6 form the set of precordial electrodes. As
was explained in chapter 2, the 12 lead ECG contains the three limb leads, three augmented
limb leads and 6 precordial leads. The potentials differences recorded by these leads can be
determined according to
VLeadI = φL − φR
(5.3)
VLeadII = φF − φR
(5.4)
VLeadIII = φF − φL
(5.5)
The signals recorded with the precordial leads follow from
Vi = φVi − φCT
for i = 1, 2, 3, 4, 5, 6
(5.6)
The three augmented limb leads are calculated with respect to the mean of the remaining
limb leads which gives for example for the aV R lead
VaVR = φR −
φF + φ L
2
(5.7)
For the aVL and aVF lead a similar expression can be derived.
5.2
Results
Surface equipotential maps of the front side of the thorax can be seen in figure 5.5. The
results will be discussed seen from the torso, which means left and right are switched from
our point of view.
Eight surface equipotential plots were made during activation. During early activation, 10
ms, the equipotential maps show a dipolar pattern, with a minimum on the left side of the
torso and a maximum slightly below the center. The maximum has a much larger amplitude
than the minimum. At 20 ms the maximum and the minimum have increased in amplitude
and a second maximum appears at the top left. During the middle phase of the depolarization, 30 to 40 ms, the minimum increases with respect to the maximum. The maximum
spreads over the left side of the thorax. During late activation, 50 to 60 ms, the minimum is
concentrated at the right side and the maximum at the left side of the torso. The amplitude
of both extremes drops in time. At the end of the depolarization phase, 70 ms, the amplitude
of the potential is around zero.
Four equipotential surface plots were made for the repolarization. At 180 ms, early depolarization, two major peaks can be seen. A minimum on the right and a maximum on the left
side. A second, much smaller maximum is visible on the left and a small minimum on the
top left. Later during repolarization, 240 to 280 ms the pattern remains virtually unchanged
but the amplitude decreases.
37
5.3 Discussion
From cardiac activation to body surface potentials
ECG signals determined in 12 leads are shown in figure 5.7. The lead signals all show an
early signal from 10 ms to 75 ms after initiation of activation, and a later peak around 200
ms after the start of activation. The first signal seems to be a typical QRS signal, and the
late signal could be a T-peak present in normal ECG recordings (see chapter 2). Lead I and
aVL show a positive QRS peak, Leads III shows a negative ORS. Lead II, V 1 , V2 and aVF
show an oscillating signal. In the signals determined in aV R V2 , V4 , V5 and V6 more clearly a
biphasic signal is present. The T-peaks of Lead I, aV L , V1 and V2 are negative, Lead II, III,
aVF , V3 and V4 show a positive T-peak. In leads aVR , V5 and V6 the T-peaks are biphasic,
aVR shows an opposite signal with respect to the other two leads. The amplitudes of most
lead signals are below 5 mV, lead V3 and V4 have an amplitude of more than 10 mV.
5.3
Discussion
In the model described in this chapter, activation times are determined using an Eikonal
equation and coupled to an action potential shape. Laplace’s equation is solved to obtain
a potential distribution throughout the torso. With this model we are able to determine
potentials on the outer surface of the torso. To assess how realistic these surface potentials
are we compare them to body surface potentials determined by Miller et al.
The surface potential maps shown in figure 5.5 differ from the surface equipotential maps
calculated by Miller et al.(see figure 5.6). They see a central maximum at 10, 20 and 30 ms
which increases from 0.25 mV to 1 mV. In our simulations also a maximum is visible in those
time steps, of which the amplitude increases from approximately 4 mV at 10 ms to 13 mV at
30 ms. We also clearly see a minimum on the left where Miller sees it on the right side.
At 35 and 40 ms the maps show some similarities, in both simulations a maximum is visible
on the left which is curled around a minimum on the right. However, when looking at the
amplitudes of the extremes we see that in our results the minimum has a two times higher
amplitude than the maximum, where in the results of Miller, the amplitude of the minimum
stays below the maximum amplitude until at 50 ms.
At 50 ms and 60 ms the results are difficult to relate, in the results of Miller a strong minimum can be seen in the center of the thorax, while in our results besides a small minimum,
a maximum is constantly present in the center. The amplitudes drop until they are close to
zero at 70 ms.
During early repolarization, 180 ms, in our results a maximum and a minimum of equal
strength are present central at the torso, while Miller sees a central maximum and a relatively small minimum at the top right. At 240 ms, in our results the amplitude of both
extremes becomes approximately 2 times higher, where in the simulations of Miller the amplitudes become more than 4 times higher. In both simulations the pattern does not change
to the end of depolarization (280 ms) while the amplitudes drop.
Also the ECG signals calculated with the model show differences with the results of Miller et
al (see figure 5.8). The QRS complex only matches for Lead I. The T tops for Leads I, aV L ,
V2 , and V6 , are opposite in sign.
In both the surface plots and the ECG simulations, the amplitude of the potential differences
is not equivalent with what Miller found. This may be caused by the fact that Miller et al.
uses an unknown scaling factor to make sure all potentials have a maximum of 2 mV. When
looking at the relation between the amplitudes our results seem to be within the same limits.
38
5.3 Discussion
From cardiac activation to body surface potentials
Since the determined potentials are relative potentials, the amplitude does not say as much
as the sign of the signal does.
Overall, the results of the present simulation are quite different from what Miller et al described. Miller et al. found similarity in their results with body surface potential measurements performed on a human subject by Taccardi [4], therefore we conclude that our
simulation is not realistic yet.
Some aspects that may be a cause of the dissimilarities between the results could be the
positioning of the heart in the thorax and the shape of the thorax itself. The shape of the
thorax used in the present simulation was simplified and although the dimensions were kept
close to the ones used in Miller et al., this could result in a different potential pattern on the
surface. In previous studies it is concluded that the geometry of the torso must be accurately
modeled in order to obtain an accurate solution [14], [8].
The orientation and position of the heart was estimated according to sketches made by Miller
et al., perhaps the heart is positioned slightly different and therefore a different potential
distribution is seen. The influence of the orientation and location of the heart in relation to
the torso was studied by Huiskamp et al. [14]. They showed the effect of small changes in
rotation and translation of the heart on the calculation of the ECG from cardiac activity. They
found that small changes in geometry can lead to changes in QRS amplitude. Furthermore
they stated that in solving the forward problem, the effect of rotation of the heart is of greater
influence on the QRS than the location.
Between the results of our simulation and the results of Miller et al, major differences were
seen in the T waves of the ECG signals which are known to result from the repolarization of
the ventricles. A possible reason for this difference could therefore be the fact that a constant
action potential duration was used throughout the heart while Miller uses a heterogeneous
action potential duration. An other factor that could influence the results is the exclusion of
the activity of the atria in our simulations. It is known that the depolarization of the atria
gives rise to the P peak, therefore, the exclusion of the atria is probably the reason why there
are no P peaks present in the ECG simulations. Since Miller did not consider the atria either,
this will probably not be a reason why our results do not match with those obtained by Miller.
However, this does make the ECG signals hard to relate to physiological measurements.
To get more insight in the influence of several aspects discussed above, in the next chapter,
a parameter validation study is performed.
39
5.3 Discussion
From cardiac activation to body surface potentials
Factor = 4.19
Factor = 10.45
Factor = 13.08
Factor = 15.7
500
500
1
500
1
500
450
450
0.8
450
0.8
450
1
1
0.8
0.8
Factor = 15.7
500
1
400
400
0.6
400
0.6
400
350
350
0.4
350
0.4
350
400 0.4
0.4
0.6
300
300
0.2
300
0.2
300
350 0.2
0.4
0.2
300
0.2
250
250
0
250
0
250
450
0.6
0.6
0.8
0
0
250
200
200
−0.2
200
−0.2
200
0
−0.2
−0.2
200
−0.2
150
150
−0.4
150
−0.4
150
100
100
−0.6
100
−0.6
100
100 −0.6
−0.6
−0.6
50
50
−0.8
50
−0.8
50
50 −0.8
−0.8
−0.8
0
−100
0
0
100
10 ms
−1
−100
0
0
100
20 ms
−1
−100
Factor = 4.96
Factor = 13.9
0
0
100
30 ms
150
−0.4
−0.4
−0.4
0
−1
−100
−100
Factor = 1.51
0
0
100
100
35 ms
−1
−1
levels
Factor = 0.069
500
500
1
500
1
500
450
450
0.8
450
0.8
450
1
1
0.8
0.8
Factor = 0.069
500
1
400
400
0.6
400
0.6
400
350
350
0.4
350
0.4
350
400 0.4
0.4
0.6
300
300
0.2
300
0.2
300
350 0.2
0.4
0.2
250
250
0
250
0
250
300
0.2
450
0.6
0.6
0.8
0
0
250
200
200
200
−0.2
−0.2
200
0
−0.2
−0.2
200
−0.2
150
150
−0.4
150
−0.4
150
100
100
−0.6
100
−0.6
100
100 −0.6
−0.6
−0.6
50
50
−0.8
50
−0.8
50
50 −0.8
−0.8
−0.8
0
−100
0
0
100
40 ms
−1
−100
Factor = 1.44
0
0
100
50 ms
−1
−100
Factor = 2.97
0
0
100
60 ms
150
−0.4
0
−1
−100
−100
Factor = 0.9387
−0.4
−0.4
0
0
100
100
70 ms
−1
−1
levels
Factor = 0.2
500
500
1
500
1
500
450
450
0.8
450
0.8
450
1
0.8
1
0.8
Factor = 0.2
500
1
400
400
0.6
400
0.6
400
350
350
0.4
350
0.4
350
400 0.4
0.4
0.6
300
300
0.2
300
0.2
300
350 0.2
0.4
0.2
300
0.2
250
250
0
250
0
250
450
0.6
0.6
0.8
0
0
250
200
200
−0.2
200
−0.2
200
0
−0.2
−0.2
200
−0.2
150
150
−0.4
150
−0.4
150
100
100
−0.6
100
−0.6
100
100 −0.6
−0.6
−0.6
50
50
−0.8
50
−0.8
50
50 −0.8
−0.8
−0.8
0
−100
0
100
180 ms
0
−1
−100
0
100
240 ms
0
−1
−100
0
100
260 ms
0
150
−0.4
−0.4
−0.4
0
−1
−100
−100
0
0
100
100
280 ms
−1
−1
levels
Figure 5.5: Equipotential lines of potentials determined on the outer surface of the torso, seen
from the front. 8 time steps during depolarization and 4 during repolarization are shown.
40
5.3 Discussion
From cardiac activation to body surface potentials
10 ms
20 ms
30 ms
35 ms
40 ms
50 ms
60 ms
70 ms
180 ms
240 ms
260 ms
280 ms
Figure 5.6: Surface equipotential lines on the front of the torso determined by Miller et al.
[31]
41
5.3 Discussion
From cardiac activation to body surface potentials
Lead I
Lead II
Lead III
aVR
aVL
aVF
v1
v2
v3
v4
v5
v6
Potential (mV)
10
5
0
0
200
400
Time (ms)
Figure 5.7: ECG determined in 12 leads on the torso surface, with the heart in the reference
position.
42
5.3 Discussion
From cardiac activation to body surface potentials
Figure 5.8: The ECG determined in 12 leads by Miller et al.[31]
43
5.3 Discussion
From cardiac activation to body surface potentials
44
Chapter 6
Parameter study
In the previous chapter a simulation was performed with the heart in a reference orientation
placed inside a torso geometry. The results were not directly resembling the results described
by Miller et al. In this chapter several adjustments are made to the settings of the model,
to investigate the influence of different parameters. First, the action potential duration was
made variable throughout the cardiac wall. Furthermore, the influence of the orientation of
the heart is studied by rotating the heart in three directions with varying angles.
6.1
6.1.1
Heterogeneous action potential duration
Methods
The action potential duration is known to vary between regions in the myocardium. In Miller
et al. the repolarization time was altered from endocardium to epicardium and from apex to
base. This could mean that later activated regions are repolarized before the early activated
regions. In the previous simulation the action potential duration was kept constant. This
means a that the repolarization sequence through the heart lasts just as long as the depolarization, approximately 60 ms. Here, a simulation is performed where the later activated
regions have an action potential duration that is 40 ms shorter than the early activated regions. This means that the duration of the repolarization sequence is decreased to 20 ms from
early activated regions to late activated regions. In a second simulation the repolarization is
made simultaneous throughout the heart by decreasing the action potential duration for late
activated regions with 60 ms with respect to the action potential duration in early activated
regions. The longest and shortest action potentials are shown in figure 6.1.
Since the repolarization of the ventricles is known to give rise to the T wave, possible effects
of this variation are expected to be visible in the T waves.
6.1.2
Results
Since the effects are expected to be visible in the T waves of the ECG signal, a 12 lead ECG
is calculated for both simulations. The results can be seen in figure 6.2 and 6.3, the ECG
determined in the previous simulation is added to serve as a reference.
45
6.1 Heterogeneous action potential duration
Parameter study
20
0
Potential (mV)
−20
−40
−60
−80
−100
0
50
100
150
Time (ms)
200
250
300
Figure 6.1: Action potential dynamics used in the simulation with the solid line representing
the longest action potential duration applied in the early activated regions, the dotted line
showing the action potential with a duration 40 ms shorter than in early activated regions
and the dashed and dotted line showing the action potential with a duration 60 ms shorter
than in early activated regions applied in the late activated regions.
Lead I
v1
Lead II
v2
Lead III
aVR
aVL
aVF
v3
v4
v5
v6
5
Potential (mV)
0
0
200
Time (ms)
400
Figure 6.2: 12 Lead ECG calculated for simulation with heterogeneous action potential duration, resulting in a repolarization sequence of 20 ms, the thin line represents the reference
simulation and the thick line represents the present simulation.
In all leads in figure 6.2 there is a change visible with respect to the reference simulation. In
all leads the amplitude of the T-wave has decreased. In figure 6.3 in all leads the effect has
increased with respect to the results in figure 6.2. In most leads, the amplitude of the T wave
is close to zero.
46
6.2 Orientation of the heart
Lead I
v1
Lead II
v2
Parameter study
Lead III
v3
aVR
aVL
aVF
v4
v5
v6
5
Potential (mV)
0
0
200
Time (ms)
400
Figure 6.3: 12 Lead ECG calculated for simulation with heterogeneous action potential duration resulting in a simultaneous repolarization, the thin line represents the reference simulation
and the thick line represents the present simulation.
6.1.3
Discussion
In both simulations the T waves of the ECG signals differ with the reference. The main change
in figure 6.2 was a decrease in amplitude. When the repolarization was made simultaneous,
the effect increases and the T wave disappears in most leads. Since the T wave reflects the
repolarization, it could be expected that with a simultaneous repolarization the T waves
disappear. A heterogeneous action potential duration clearly affects the T waves in the ECG,
therefore it should be considered to incorporate this in the model. However, the ECG signals
still are not yet realistic, this means that this parameter needs more study.
6.2
Orientation of the heart
To study the influence of the orientation of the heart, the heart is rotated in steps around the
x, y and z axes with respect to the torso.
6.2.1
Methods
Rotations of π/18 radians are performed clockwise and counter clockwise around the x, y and
z axis with respect to the torso. This means that 6 additional simulations are performed.
47
6.2 Orientation of the heart
6.2.2
Parameter study
Results
In figure 6.4 equipotential maps are shown in three time-steps during depolarization for rotation of π/18 rad around the x axis. This result is shown since it produces the largest change
with respect to the reference simulation. The results of the other simulations can be seen
in appendix 7. One time step during early activation is shown (10ms), one in the middle of
the depolarization, 40 ms, and one at the end of the depolarization phase at 70 ms. The
equipotential maps of the reference simulations are added for comparison.
Factor = 1.56
Factor = 15.92
Factor==0.06
0.06
Factor
500
500
1
500
500
1
450
450
0.8
450
0.8
0.8
0.8
400
400
0.6
400
0.6
0.6
0.6
350
350
0.4
350
0.4
0.4
0.4
300
300
0.2
300
0.2
0.2
0.2
250
250
0
250
0
00
200
200
−0.2
200
200
−0.2
−0.2
−0.2
150
150
−0.4
150
150
−0.4
−0.4
−0.4
100
100
−0.6
100
100
−0.6
−0.6
−0.6
50
50
−0.8
50
50
−0.8
−0.8
−0.8
−1
−100
0
0
−1
−100
−100
0
−100
0
100
0
10 ms
Factor = 4.19
0
100
40 ms
Factor = 13.9
0
0
11
100
100
70 ms
Factor==0.069
0.069
Factor
−1
−1
levels
500
500
1
500
500
1
450
450
0.8
450
0.8
0.8
0.8
400
400
0.6
400
0.6
0.6
0.6
350
350
0.4
350
0.4
0.4
0.4
300
300
0.2
300
0.2
0.2
0.2
250
250
0
250
0
00
200
200
−0.2
200
200
−0.2
−0.2
−0.2
150
150
−0.4
150
150
−0.4
−0.4
−0.4
100
100
−0.6
100
100
−0.6
−0.6
−0.6
50
50
−0.8
50
50
−0.8
−0.8
−0.8
0
−1
−100
0
0
−1
−100
−100
0
−100
0
100
10 ms
0
100
40 ms
0
0
11
100
100
70 ms
−1
−1
levels
Figure 6.4: Surface equipotential lines on the front side of the torso. Top: Results of the
simulation with the heart rotated 10 degrees around the x axis in three time steps after the
start of depolarization. Bottom: results of the reference simulation.
48
6.2 Orientation of the heart
Parameter study
In the results in figure 6.4 the main difference with the reference simulation is visible at 10
ms. Here the amplitude is lower and the minimum has a higher amplitude in relation to the
maximum than in the reference. At 40 ms the amplitude in the present simulation is higher
but the pattern is similar. At the end of depolarization (70 ms) the amplitude is close to zero,
similar as in the results of the reference simulation.
The ECG signals calculated from a simulation with the heart rotated π/18 radians around
the x axis is shown in figure 6.5.
Lead I
Lead II
Lead III
aVR
aVL
aVF
v1
v2
v3
v4
v5
v6
5
Potential (mV)
0
0
200
Time (ms)
400
Figure 6.5: 12 lead ECG calculated from potentials determined with the heart rotated π/18
rad around the x axis. The thin line represents the reference signal the thick line shows the
result of the present simulation.
In the ECG signals the main differences of the simulation with the reference can be seen in
the V1 and V2 leads, where the T-wave has changed sign. In the other leads no or not much
difference between both simulations can be found.
6.2.3
Discussion
The result of only one simulation is described here, because the other simulations do not
provide extra information. Huiskamp et al. found that the main change in the ECG after a
49
6.2 Orientation of the heart
Parameter study
rotation of 0.03π resulted in amplitude changes in the QRS complexes in the V 2 and V3 lead
[14]. Our results also show amplitude changes in the precordial leads, however mainly the T
waves are affected. Since the results presented here are the results of the simulation with the
most influence, it can be concluded that the rotation of the heart is not the most important
factor influencing the resulting body surface potentials.
50
Chapter 7
General discussion
The aim of this study was to develop and implement a model to solve the forward problem
of electrocardiography.
A model is presented that is capable of producing body surface potentials from a representation of cardiac electrical activity. A 12 lead ECG can be calculated which shows a QRS peak
and a T wave that are typically present in clinical ECG measurements. The results obtained
with the model are compared to results obtained by Miller et al. who used a multiple dipole
approach to simulate body surface potentials. Reproducing the body surface potentials and
ECG’s from Miller et al. seemed not achievable with this model.
Some parameters that could be of influence on the body surface potentials were further analyzed. Heterogeneity of the action potential duration throughout the cardiac wall was found
to have an effect on the T-waves only in the ECG determined in all 12 leads.
The effect of the orientation of the heart inside the torso was studied by rotating the heart in
various directions. The rotations over an angle of π/18 rad had a minor effect on the ECG’s
calculated with the model.
None of the variations resulted in an improved agreement with the results of Miller et al.,
this means improvement of the model is needed to be able to obtain more realistic results.
Improvements of the model could be made at various points. First of all, a heterogeneous
action potential duration throughout the cardiac tissue should be incorporated in the model.
The distribution of the heterogeneity however, has to be studied in more detail.
Furthermore the geometry of the torso could have an effect on the body surface potential
pattern. The geometry of the torso used in this study is highly simplified. Studies have
shown that the geometry of both heart and torso should be modeled with sufficient accuracy
to obtain an accurate forward solution [14]. This means that if we want to simulate realistic
body surface potentials, realistic geometries should be used. Since rotation of the heart did
not have major effects on the ECG, it is not expected that this will be a solution for the
difficulties encountered in reproducing the ECG’s of Miller.
An other improvement that could be made in the model is in the amount of nodes used to
describe the cardiac wall. Since the activation of the heart is translated from an activation
described on 37405 nodes in the activation model to 1430 nodes in the present model, there
could be deviations in the activation sequence. To validate the effect of mesh resolution on
51
General discussion
the activation pattern, further analysis of this parameter is needed. To make the model more
realistic it could be considered to incorporate torso inhomogeneities into the model. It is
known that especially the low conductivity of the lungs has a major effect on the smoothing
of the body surface potentials [28], [6], [3]. The use of finite elements makes it relatively easy
to incorporate inhomogeneities when the geometry of the concerning region is known.
Despite the fact that results of Miller et al. could not be reproduced with the model, it
was shown that our model is capable of producing ECG-like signals from a representation of
cardiac activation. Additional analysis of parameters influencing the solution calculated with
the model is needed to eventually be able to produce realistic body surface potential maps
and ECG’s.
After that a next step would be to model an inverse analysis of the ECG. When a cardiac
activation sequence can be determined from the ECG using a mathematical model, this could
be incorporated in models of cardiac mechanics, which would be a step towards patient specific
modeling.
52
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56
Appendix A
Results of variations in orientation
of the heart.
Below, the results belonging to the simulations described in section 6.2 are shown.
Lead I
Lead II
Lead III
aVR
aVL
aVF
v1
v2
v3
v4
v5
v6
5
Potential (mV)
0
0
200
Time (ms)
400
Figure A.1: 12 lead ECG signal determined with the heart rotated π/18 rad counterclockwise
around the x axis. The thin line represents the reference simulation, the thick line the present
simulation.
57
Results of variations in orientation of the heart.
Factor = 14.5
Factor = 2.86
Factor==0.08
0.08
Factor
500
500
1
500
500
1
450
450
0.8
450
0.8
0.8
0.8
400
400
0.6
400
0.6
0.6
0.6
350
350
0.4
350
0.4
0.4
0.4
300
300
0.2
300
0.2
0.2
0.2
250
250
0
250
0
00
200
200
-0.2
200
200
−0.2
−0.2
−0.2
150
150
-0.4
150
150
−0.4
−0.4
−0.4
100
100
-0.6
100
100
−0.6
−0.6
−0.6
50
50
-0.8
50
50
−0.8
−0.8
−0.8
0
0
−1
−100
−100
0
-100
0
100
10 ms
0
-1
−100
0
100
40 ms
0
0
11
100
100
70 ms
−1
−1
levels
Figure A.2: Surface equipotential lines on the front side of the torso with the heart rotated
π/18 rad counterclockwise around the x axis. Three time steps after the start of depolarization
are shown.
58
Results of variations in orientation of the heart.
Lead I
v1
Lead II
v2
Lead III
v3
aVR
aVL
aVF
v4
v5
v6
5
Potential (mV)
0
0
200
Time (ms)
Lead I
Lead II
Lead III
aVR
aVL
aVF
v1
v2
v3
v4
v5
v6
400
5
Potential (mV)
0
0
200
Time (ms)
400
Figure A.3: 12 lead ECG signal determined with the heart rotated clockwise (top) and counterclockwise (bottom) around the y axis. The thin line represents the reference simulation,
the thick lines the present simulations.
59
Results of variations in orientation of the heart.
Factor = 1.64
Factor = 17
Factor==0.068
0.068
Factor
500
500
1
500
500
1
11
450
450
0.8
450
0.8
0.8
0.8
400
400
0.6
400
0.6
0.6
0.6
350
350
0.4
350
0.4
0.4
0.4
300
300
0.2
300
0.2
0.2
0.2
0
00
250
250
0
250
200
200
−0.2
200
200
−0.2
−0.2
−0.2
150
150
−0.4
150
150
−0.4
−0.4
−0.4
100
100
−0.6
100
100
−0.6
−0.6
−0.6
50
50
−0.8
50
50
−0.8
−0.8
−0.8
0
−1
−100
0
0
−1
−100
−100
0
−100
0
100
10 ms
Factor = 3.27
0
100
20 ms
Factor = 15.77
0
0
100
100
30 ms
Factor==0.07
0.07
Factor
−1
−1
levels
500
500
1
500
500
1
450
450
0.8
450
0.8
0.8
0.8
400
400
0.6
400
0.6
0.6
0.6
350
350
0.4
350
0.4
0.4
0.4
300
300
0.2
300
0.2
0.2
0.2
250
250
0
250
0
00
200
200
−0.2
200
200
−0.2
−0.2
−0.2
150
150
−0.4
150
150
−0.4
−0.4
−0.4
100
100
−0.6
100
100
−0.6
−0.6
−0.6
50
50
−0.8
50
50
−0.8
−0.8
−0.8
0
−1
−100
0
0
−1
−100
−100
0
−100
0
100
10 ms
0
100
20 ms
0
0
11
100
100
30 ms
−1
−1
levels
Figure A.4: Surface equipotential lines on the front side of the torso with the heart rotated
π/18 rad clockwise (top) and counterclockwise (bottom) around the y axis. Three time steps
after the start of depolarization are shown.
60
Results of variations in orientation of the heart.
Lead I
Lead II
Lead III
aVR
aVL
aVF
v1
v2
v3
v4
v5
v6
5
Potential (mV)
0
0
200
Time (ms)
Lead I
Lead II
Lead III
aVR
aVL
aVF
v1
v2
v3
v4
v5
v6
400
5
(mV)
0
0
200
Time (ms)
400
Figure A.5: 12 lead ECG signal determined with the heart rotated clockwise (top) and counterclockwise (bottom) around the z axis. The thin line represents the reference simulation,
the thick lines the present simulations.
61
Results of variations in orientation of the heart.
Factor = 4.02
Factor = 11.85
Factor==0.09
0.09
Factor
500
500
1
500
500
1
11
450
450
0.8
450
0.8
0.8
0.8
400
400
0.6
400
0.6
0.6
0.6
350
350
0.4
350
0.4
0.4
0.4
300
300
0.2
300
0.2
0.2
0.2
0
00
250
250
0
250
200
200
−0.2
200
200
−0.2
−0.2
−0.2
150
150
−0.4
150
150
−0.4
−0.4
−0.4
100
100
−0.6
100
100
−0.6
−0.6
−0.6
50
50
−0.8
50
50
−0.8
−0.8
−0.8
0
−1
−100
0
0
−1
−100
−100
0
−100
0
100
10 ms
Factor = 0.98
0
100
40 ms
Factor = 13.56
0
0
100
100
70 ms
Factor==0.055
0.055
Factor
−1
−1
levels
500
500
1
500
500
1
450
450
0.8
450
0.8
0.8
0.8
400
400
0.6
400
0.6
0.6
0.6
350
350
0.4
350
0.4
0.4
0.4
300
300
0.2
300
0.2
0.2
0.2
250
250
0
250
0
00
200
200
−0.2
200
200
−0.2
−0.2
−0.2
150
150
−0.4
150
150
−0.4
−0.4
−0.4
100
100
−0.6
100
100
−0.6
−0.6
−0.6
50
50
−0.8
50
50
−0.8
−0.8
−0.8
0
−1
−100
0
0
−1
−100
−100
0
−100
0
100
10 ms
0
100
40 ms
0
0
11
100
100
70 ms
−1
−1
levels
Figure A.6: Surface equipotential lines on the front side of the torso with the heart rotated
π/18 rad clockwise (top) and counterclockwise (bottom) around the z axis. Three time steps
after the start of depolarization are shown.
62